 reserve V for Z_Module;
 reserve W for Subspace of V;
 reserve v, u for Vector of V;
 reserve i for Element of INT.Ring;

theorem ThTF3C:
  for R being Ring
  for V, W being LeftMod of R,
  T be linear-transformation of V,W,
  A being Subset of V
  holds T.:(the carrier of Lin(A)) c= the carrier of Lin(T.: A)
  proof
    let R be Ring;
    let V, W be LeftMod of R,
    T be linear-transformation of V,W,
    A be Subset of V;
    for y be object st y in T.:(the carrier of Lin(A)) holds
    y in the carrier of Lin(T.:A)
    proof
      let y be object;
      assume y in T.:(the carrier of Lin(A));
      then consider x be Element of V such that
      A2: x in the carrier of Lin(A) & y = T.x by FUNCT_2:65;
      x in Lin(A) by A2;
      then consider l be Linear_Combination of A such that
      A3: x = Sum(l) by MOD_3:4;
      reconsider l as Linear_Combination of V;
      reconsider Tl = T @* l as
      Linear_Combination of T.:(Carrier l) by ZMODUL05:44;
      Sum(Tl) = T.(Sum l) by ZMODUL05:46; then
      A5: y in Lin(T.:Carrier l) by A2,A3,MOD_3:4;
      T .: (Carrier l) c= T.: A by RELAT_1:123,VECTSP_6:def 4;
      then Lin(T.:Carrier l) is Subspace of Lin(T.:A) by MOD_3:10;
      then the carrier of Lin(T.:Carrier l)
      c= the carrier of Lin(T.:A) by VECTSP_4:def 2;
      hence y in the carrier of Lin(T.:A) by A5;
    end;
    hence thesis;
  end;
